Relative maxima and minina

Definition

Given a function f (x , y) of two variables,

We say that f has a local maximum at the point (a, b) if

f (x , y) f (a, b)

for all (x , y) close enough to (a, b).

We say that f has a local minimum at the point (a, b) if

f (x , y) f (a, b)

for all (x , y) close enough to (a, b).

Todays goal: Given a function f , identify its local maxima and minima.

Math 105 (Section 203) Multivariable Calculus Extremization 2010W T2 1 / 6

Relative maxima and minina

Definition

Given a function f (x , y) of two variables,

We say that f has a local maximum at the point (a, b) if

f (x , y) f (a, b)

for all (x , y) close enough to (a, b).

We say that f has a local minimum at the point (a, b) if

f (x , y) f (a, b)

for all (x , y) close enough to (a, b).

Todays goal: Given a function f , identify its local maxima and minima.

Math 105 (Section 203) Multivariable Calculus Extremization 2010W T2 1 / 6

The first derivative test

Description of the test

The first step in finding local max or min of a function f (x , y) is to findpoints (a, b) that satisfy the two equations

fx(a, b) = 0 and fy (a, b) = 0.

Any such point (a, b) is called a critical point of f .

Note: Any local max or min of f has to be a critical point, but everycritical point need not be a local max or min.

Math 105 (Section 203) Multivariable Calculus Extremization 2010W T2 2 / 6

The first derivative test

Description of the test

The first step in finding local max or min of a function f (x , y) is to findpoints (a, b) that satisfy the two equations

fx(a, b) = 0 and fy (a, b) = 0.

Any such point (a, b) is called a critical point of f .

Note: Any local max or min of f has to be a critical point, but everycritical point need not be a local max or min.

Math 105 (Section 203) Multivariable Calculus Extremization 2010W T2 2 / 6

Finding critical points : an example

Find all critical points of the following function

f (x , y) =1

x+

1

y+ xy .

A. (0, 0), (1, 1)

B. (1, 1)

C. (0, 0), (1, 1), (1,1), (1, 1), (1,1)D. There is no critical point

E. (0, 0)

Math 105 (Section 203) Multivariable Calculus Extremization 2010W T2 3 / 6

The previous example (ctd)

Is the critical point (1, 1) a local max, a local min or neither?

The second derivative test

If (a, b) is a critical point of f , calculate D(a, b), where

D = fxx fyy f 2xy .

1. If D(a, b) > 0 and fxx(a, b) < 0, then f has a local maximum value at(a, b).

2. If D(a, b) > 0 and fxx(a, b) > 0, then f has a local minimum value at(a, b).

3. If D(a, b) < 0, then f has a saddle point at (a, b).

4. If D(a, b) = 0, then the test is inconclusive.

Math 105 (Section 203) Multivariable Calculus Extremization 2010W T2 4 / 6

Classifying critical points : an example

In the example

f (x , y) =1

x+

1

y+ xy

determine whether the critical point (1, 1) is

A. a local minimum

B. a local maximum

C. a saddle point

D. neither of the above

Math 105 (Section 203) Multivariable Calculus Extremization 2010W T2 5 / 6

An application

A company manufactures two products A and B that sell for $10 and $9per unit respectively. The cost of producing x units of A and y units of Bis

400 + 2x + 3y + 0.01(3x2 + xy + 3y2).

Find the values of x and y that maximize the companys profits.

A. (100, 80)

B. (120, 90)

C. (120, 80)

D. (80, 120)

Math 105 (Section 203) Multivariable Calculus Extremization 2010W T2 6 / 6